The Structure of Quantum States
The quantum state is the fundamental entity in quantum mechanics. This chapter develops the basic mathematical structure of quantum states by developing the correspondence between states and ordinary vectors. The familiar scalar or dot product of vectors is used to develop the concepts of projection, orthonormality, linear independence, and expansion over a complete set of basis vectors. These ideas are then extended from vectors in a vector space to functions in a function space. Dirac's bra-ket notation is introduced, and the concepts developed for ordinary vectors are revisited in the context of quantum states and Dirac notation. The central quantum concept of a representation is then explored, and the relation between representations and superpositions is illustrated with the infinite potential well. The final topic is representational freedom: the fact that quantum mechanics remains valid regardless of the representation chosen for calculation.
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